To solve this problem, we need to understand the relationship between the atomic number $ Z $, the characteristic X-ray wavelengths, and the cut-off wavelength in X-ray spectra.

The cut-off wavelength, also known as the minimum wavelength ($ \lambda_{\text{min}} $), corresponds to the maximum energy of the X-rays produced and can be related to the accelerating voltage used in the X-ray tube. The characteristic X-rays, such as the $ K_\alpha $-line, correspond to specific electron transitions in the atom and are dependent on the atomic number $ Z $ of the target material.

The wavelength of the $ K_\alpha $-line is given by Moseley's Law, which can be approximated as:

$$ \lambda_{K_\alpha} \propto \frac{1}{(Z - 1)^2} $$

The cut-off wavelength is given by the relationship between the energy of the incident electrons and the X-ray produced:

$$ \lambda_{\text{min}} = \frac{hc}{eV} $$

Now, given the ratio $ r $ for the first metal target with $ Z = 46 $ :

$$ r = \frac{\lambda_{K_\alpha}}{\lambda_{\text{min}}} = 2 $$

For the second metal target with $ Z = 41 $, let's denote the new ratio as $ r_2 $. We assume the same cutoff wavelength since the same electron beam is used.

The ratio for $ Z = 41 $ can be calculated using the relationship between the $ K_\alpha $-line wavelenghts:

$$ \frac{\lambda_{K_\alpha (1)}}{\lambda_{K_\alpha (2)}} = \left( \frac{Z_2 - 1}{Z_1 - 1} \right)^2 $$

Substituting $ Z_1 = 46 $ and $ Z_2 = 41 $:

$$ \frac{\lambda_{K_\alpha (1)}}{\lambda_{K_\alpha (2)}} = \left( \frac{41 - 1}{46 - 1} \right)^2 = \left( \frac{40}{45} \right)^2 = \left( \frac{8}{9} \right)^2 = \frac{64}{81} $$

Let $ \lambda_{K_\alpha (1)} = 2 \lambda_{\text{min}} $ because $ r = 2 $. Therefore:

$$ \lambda_{K_\alpha (2)} = \lambda_{K_\alpha (1)} \cdot \frac{81}{64} = 2 \lambda_{\text{min}} \cdot \frac{81}{64} $$

Thus, the new ratio $ r_2 $ is:

$$ r_2 = \frac{\lambda_{K_\alpha (2)}}{\lambda_{\text{min}}} = 2 \cdot \frac{81}{64} = \frac{162}{64} = 2.53 $$

So, the value of $ r $ for the second metal target with $ Z = 41 $ is:

Option A: 2.53